---
tag: 6W69H3A3G5
kind: concept
name: Markov random field
msc: [62H22, 60G60]
specializes: [Q3FQAD1BXG]
sources:
  - url: https://en.wikipedia.org/wiki/Markov_random_field
same_as:
  wikidata: Q176827
  wikipedia: Markov random field
---

## Summary

A Markov random field is a family of random variables indexed by the vertices of an undirected [[graph|00C0BR7D7H]].
Each variable depends on the others only through its neighbours.
It is the undirected kind of [[graphical model|Q3FQAD1BXG]].
By the [[Hammersley–Clifford theorem|K07QH8EKD0]], positive Markov random fields are exactly the distributions that factor over the [[cliques|0BYX1D7C2E]] of the graph.
The Ising model and many models in image analysis are Markov random fields.

## Explanations

### L1 human

Think of the pixels of a black and white image, each joined to its four neighbours.
In a Markov random field, once the colours of the neighbours of a pixel are known, the rest of the image tells nothing more about that pixel.
This captures the idea that nearby pixels tend to agree while far pixels interact only through the pixels between them.
The same picture works for spins in a magnet, which is the Ising model.